Method for analyzing structure having deformable rigid elements

ABSTRACT

A method of analyzing a structure containing a rigid body. The method defines a set of displacement factors to be imposed on a structure model. The method is used for calculating structure stress defining equations including a load vector representing externally applied forces on the structure model. The method modifies the load vector to compensate for rigid body deformation and evaluates stresses on the structure model from data generated from the structure stress defining equations.

This is a continuation of U.S. patent application Ser. No. 08/1723,104,filed Oct. 1, 1996, abandoned.

BACKGROUND OF THE INVENTION

The present invention relates generally to structural analysistechniques, and more particularly to a finite element analysis methodthat compensates for expansion of structural rigid elements due tothermal, hydraulic, mechanical or other causes of structuraldeformation.

Analysis of stresses placed on structures is an important element instructural design and testing. For example, the on-orbit and descentphases of a spacecraft often produce large structural temperaturegradients, which result in thermal stresses on the spacecraft structure.Structural stress data also result from aerodynamics, engine thrust, andvibrations. Commercially available finite element analysis programsmodel the structure and process the various types of load to evaluatethe various stresses placed thereon.

The recovery of the structural stress data is often complicated by thepresence of multi-point constraints, or rigid elements, located betweennon-coincident grid points on the model of the structure. Such rigid eltypically stiff bodies to which other elastic structural components areattached, and are usually employed as a device for avoiding excessivelylarge terms in the stiffness matrix causing it to become "illconditioned". Conventional finite element coded software programstypically have no capability for compensating for expansion of thestructural rigid elements. This lack of compensation introducesartificial constraints into the computer model, thereby causingincorrect structural stresses to be calculated.

A present solution for correcting the above limitation requiresreplacing the structural rigid elements in the computer generated modelwith artificially stiff elastic elements having appropriate expansioncoefficients and associated defined variables. However, this solution isundesirable as it results in less-than-optimal conditioning of thestiffness matrix generated by t program. In addition, replacement of therigid elements with artificially stiff bar elements increases the timerequired to generate the computer model.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a system for analyzing characteristics of astructure having rigid elements;

FIG. 2 is a first structure modeled and analyzed by the system of FIG.1;

FIG. 3 illustrates the first structure installed in a space shuttle;

FIG. 4 is a second structure modeled and analyzed by the system of FIG.1;

FIG. 5 is a third structure modeled and analyzed by the system of FIG.1;

FIG. 6 is a fourth structure modeled and analyzed by the system of FIG.1; and

FIG. 7 is a flow diagram of the methodology programmed into the systemof FIG. 1 for analyzing structural characteristics of structures such asthose shown in FIGS. 3-6.

SUMMARY OF THE INVENTION

The present invention contemplates a method of analyzing the structureof an elastic body containing deformable rigid elements. The method ofthe present invention provides a more accurate way to account for theexpansion of rigid elements in the structure by modifying presentstructural analysis finite element coded software programs, therebypreventing introduction of artificially high stiffness terms into themodel.

In particular, the present invention provides a method of analyzing thecharacteristics of a structure having a rigid element and includes thestep of defining a set of displacement factors to be imposed on astructure model. The method also includes the step of modifying therecovery of displacements to compensate for rigid element deformation.In addition, the method also includes the step of calculating a loadvector representing externally applied forces on the structure model.The method modifies the load vector to compensate for rigid elementdeformation before calculating stresses on the structure model throughuse of the load vector.

The present invention also provides a system for analyzingcharacteristics of a structure model having a rigid element thatincludes a data input for inputting rigid element displacement data intothe system. The system also includes a memory that stores structuralstress equations. A processor is provided that calculates structuralstress utilizing the above stress equations and displacement data. Thedata output outputs this data, with the data including compensation fordeformation of the rigid elements.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

Referring to FIG. 1, a system according to a preferred embodiment of thepresent invention is shown generally at 10. The system 10 is programmedto analyze characteristics of a structural model, shown generally at 12,having deformable rigid elements for structure design and modelingpurposes. In this context, the word "rigid" denotes an inelastic andnormally inextensible element whose deformation is prescribed in purelymathematical terms in the method of the present invention.

Still referring to FIG. 1, a processor 16, which is preferably aconventional personal computer including a Pentium® processor and 32megabytes of random access memory (RAM) is shown generally at 20. Theprocessor 16 is programmed with a conventional finite element codedsoftware program such as Nastran, CSA/Nastran, Abaqus, or Stardynestructural analysis and modeling as will be discussed below. Suchcommercially available software programs are coded to analyze structuralmodels to determine the various effects of expansion due to thermal,hydraulic, mechanical or other stress related phenomena on the structureitself. Additional commands including programming changes necessary toimplement the methodology of the present invention may be entered intothe processor through the operator interface 26, which is typically acomputer keyboard. Data generated by the processor through the finitecoded element analysis program is output through the data output 30.

While the system of the present invention is described generally aboveas being implemented in a conventional personal computer having aPentium® processor, it should be appreciated that other computerhardware may also utilized, such as a SUN Workstation, CRAYSupercomputer, or any other computer hardware having the processingcapability necessary to run commercially available finite element codedprograms such as those mentioned above.

Referring to FIGS. 2 and 3, a cargo element of the type that is analyzedby the system 10 is shown generally at 40. The cargo element is of thetype installed within the cargo bay of a space shuttle 41, the walls ofwhich are shown generally at 42a-c in FIG. 2, for carrying variouspayloads, such as the payload 43, on shuttle missions. The cargo elementincludes a keel 44 including fittings 46 that latch into the bottom wallof the cargo bay 42a of the shuttle. The fittings are designed to slideup and down in a mounting hole (not shown) in the bottom of the cargobay. The cargo element also includes trunnions 50 that lock into thesides 42b, 42c of the shuttle for mounting the cargo element to theshuttle. The trunnions 50 introduce a preload into the overall shuttlestructure, as the trunnions must be forced into mounting holes (notshown) in the shuttle walls in a friction fit, due to the inherentmisalignment of the mounting bores with the trunnions. Therefore, thetrunnions must be forced into the mounting bores during installation. Adisplacement factor represented by δ_(z) is shown at 54 to reflect theconstraint of the trunnion within the mounting bores in either the plusor minus Z directions.

Present commercially available finite element coded software programscompensate for deformation of the trunnion by requiring that the shuttlewalls 42a-c be modeled as an infinitely stiff shuttle when a load isplaced on the cargo element 40 in a particular direction, such as thatrepresented by δ_(z) at 54. Therefore, the accuracy of the programanalysis would be somewhat compromised. As will now be described, thepresent invention modifies existing analysis programs to enforcerelative displacements between various points in the model of the cargoelement, rather than enforcing displacements between the cargo elementand ground.

Referring to Table I below, definitions of various terms used inequations to follow are given below in Table I:

TABLE I

R_(g) !--multi-point constraint (MPC) coefficients

These are simply coefficients of the linear constraint equations and aredefined by the user to achieve desired results.

{u_(g) }--g-set displacements. The set of all displacements in thestructural model. The "g-set" is a finite element code term that simplymeans everything.

{δ_(G) }=user defined displacements

These are the imposed displacements that compensate for rigid elementdeformation due to mechanical, hydraulic, thermal or other structuralstress-related forces.

K!--Stiffness matrix

This is formed by assembling the stiffness matrices of all of the finiteelements comprising the structure model.

{P}--Load vector

This represents any externally applied loads (forces) on the structuremodel.

{q_(m) }--Vector of constraint forces on {u_(m) }

These are forces internal to the structure model resulting from theapplied loads.

The theory behind the structural analysis method of the presentinvention involves a modification of multi-point constraint equationsthat relate displacement between certain points in the structure such asthose used in structural analysis software programs such as NASTRAN. Theconstraint equations are shown below in Equation 1:

     R.sub.g !{u.sub.g }=0                                     (1)

Where:

R_(g) !--MPC constraint coefficients

{u_(g) }--g-set displacements

The modification of the above equations according to the presentinvention allows for expansion of rigid elements, such as the spaceshuttle structure 42 shown in FIG. 2, so that Equation 1 above ismodified as shown in Equation 2:

     R.sub.g !{u.sub.g }={δ.sub.G }                      (2)

Where:

{δ_(G) }=user defined displacement

As each displacement point above has an associated independent (n-set)or dependent (m-set) degree of freedom (DOF) associated therewith,Equation 2 may be partitioned into n-set and m-set degrees of freedom asshown below in Equation 3: ##EQU1##

Equation 3 can thereby be solved for the dependent m-set displacementsas shown below in Equation 4:

    {u.sub.m }= R.sub.m !.sup.-1 {δ.sub.G }+ G.sub.m !{u.sub.n }

Where:

     G.sub.m !=- R.sub.m !.sup.-1  R.sub.n !                   (4)

Where:

R_(m) !⁻¹ is the inverse of the partition of the dependent multi-pointconstraint coefficient matrix

{δ_(G) } is the user defined displacement term

R_(n) !=is the partition of the multi-point constraint coefficientmatrix corresponding to the dependent set

{u_(n) } is the independent set of displacements.

Equation 4 is identical to a standard n-set displacement recoveryequation, with the addition of the δ_(G) term to compensate forexpansion of rigid elements. As independent displacements are functionalto the structure itself, Equation 4 is used to recover dependentdisplacements from the independent displacements.

At this point, the structural problem may be written as shown inEquation 5: ##EQU2##

In Equation 5 above, each of the four terms in the stiffness matrixrepresents a partition of the stiffness matrix, with K_(nn) representinga partition of the stiffness matrix containing the independent degreesof freedom, K_(nm) representing the couplings between the dependent andindependent degrees of freedom, K_(nm) ^(T) representing the transposeof K_(nm), and K_(mm) representing the partition of the global stiffnessmatrix containing dependent degrees of freedom. Therefore, Equation 5equates to: stiffness×displacement=force. The bars over the symbolsabove in Equation 5 indicate that these particular arrays will bereplaced in the reduction process.

As the multi-point constraint Equation 4 is incorporated into Equation5, Equation 6 is derived as shown below: ##EQU3##

Where:

{Y_(m) }=- R_(m) !⁻¹ {δ_(G) }

{q_(m) ^(m) }--Vector of constraint forces on {u_(m) }

Subsequently, q_(m) ^(m) and u_(m) are eliminated to transform theequation into the same form as Equation 5. The elimination of theseterms results in the following Equation 7 given below:

     K.sub.nm +K.sub.nm G.sub.m +G.sub.m.sup.T K.sub.nm.sup.T +G.sub.m.sup.T K.sub.mm G.sub.m !{U.sub.n }={P.sub.n }+ G.sub.m.sup.T !{P.sub.m }+ K.sub.nm !{Y.sub.m }+ G.sub.m.sup.T! K.sub.mm !{Y.sub.m }(7)

From Equation 7, a reduced stiffness matrix and load vector may bederived, as shown below in Equations 8 and 9 respectively:

     K.sub.nn != K.sub.nm +K.sub.nm G.sub.m +G.sub.m.sup.T K.sub.nm.sup.T +G.sub.m.sup.T K.sub.mm G.sub.m !                         (8)

And,

     P.sub.n !={P.sub.n }+ G.sub.m.sup.T !{P.sub.m }+ K.sub.nm !{Y.sub.m }+ G.sub.m.sup.T ! K.sub.mm !{Y.sub.m }                   (9)

P_(n) ! is the load vector reduced to the independent set

{Pn}is the partition of the load vector corresponding to the independentset

G_(m) ^(T) ! is the transpose of G_(m) ! as defined in Equation 4.

{P_(m) } is the partition of the load vector corresponding to thedependent set

K_(nm) ! represents the couplings between the independent and dependentdegrees of freedom

{Y_(m) } is defined in Equation 6.

K_(mm) ! is the partition of the global stiffness matrix pertaining todependent degrees of freedom

Equation 8 is identical to the equation presented in the commerciallyavailable COSMIC/NASTRAN theoretical manual. However, Equation 9contains two additional terms involving Y_(m), to compensate forexpansion or other deformation of the rigid structure. When δ_(G) iszero, Equations 4 and 9 above degenerate to the standard equations withno expansion/deformation compensation being introduced.

It is contemplated that the above programming modifications be made toexisting software packages through programming modifications made inFORTRAN, or any other conventional computer programming language used toimplement the aforementioned programs, in a manner well known to thoseskilled in the art.

FIGS. 4 through 6 illustrate structures capable of being modeled andevaluated by finite element coded software programs such as thosementioned above, as modified by the structural analysis method of thepresent invention.

FIG. 4 illustrates a structure, shown generally at 70, including a door72 having a turnbuckle 74 connected to opposing diagonal ends 76, 78 ofthe door. Conventional finite element coded software modeling programswill enforce displacements relative to ground at each of the corners ofthe door 72 rather than applying enforced displacement relative to eachof the corners of the door 72. The system and method of the presentinvention enforce relative displacement between the corners 76, 78,thereby improving structural modeling accuracy.

FIG. 5 shows a system 80 including a structure 82 having a constrictingbelt 84 cinched around the structure capable of being variably tightenedaround the structure. Rather than determining overall displacement beingenforced around the structure, as with conventional modeling programs,the present invention allows specification of the amount of contractionon each of the four sides of the structure. Therefore, if the structureis not uniformly stiff around its perimeter, the present inventionallows the program operator to calculate the different stresses on eachof the four sides, thereby resulting in more accurate structure modelingand analysis.

FIG. 6 illustrates a system generally at 90 including a building 92being jacked from an elastic foundation 94 by a hydraulic jack 96.Conventional modeling equations would model the foundation as beinginfinitely stiff. However, the present invention, through theintroduction of the δ_(G) term, factors the inherent elasticity of thefoundation into the equation, thereby allowing the forced displacementbetween two elastic points to be applied to the model. This increasesaccuracy of the system modeling and analysis.

FIG. 7 is a flow diagram shown at 100 illustrating the methodology ofthe present invention. At step 102, a model of the structure beinganalyzed is generated. At step 104, the user defines the rigid elementdeformations δ_(G). At step 106, the system determines if the systemuser has defined displacements (δ_(G)) to compensate for rigid elementdeformation in the structure due to mechanical, hydraulic, thermal orother structure stress related forces. If no displacements are defined,the program proceeds to step 108, and does not factor in the rigidelement deformation. If the system user has defined such displacements,at step 110, u_(g) is calculated, including correction for the imposeddisplacements δ_(G). After u_(g) has been calculated, the modifiedstress defining equations are subsequently calculated at step 112. Thecorrected stress defining equations include the load vector P and themodified displacements included in u_(g). At step 114, the data isevaluated using the stress defining equations calculated at step 112.After this data is evaluated at step 116, it is used for structuralmodeling and analysis purposes as is known in the art.

Upon reading the foregoing detailed description, it should beappreciated that the system and method of the present invention enableone skilled in the art to modify an existing finite element codedstructural analysis program to compensate for deformation of structuralrigid elements. Thus, the present invention permits more accuratecomputer modeling of structures with only minimal software programmingmodification.

While the above detailed description describes the preferred embodimentof the present invention, the invention is susceptible to modification,variation and alteration without deviating from the scope and fairmeaning of the subjoined claims.

What is claimed is:
 1. A system for analyzing characteristics of astructure having a rigid element, comprising:a data input for inputtingstructure displacement data; a memory for storing a plurality ofstructure stress defining equations adjusted to compensate fordeformation of the rigid element, wherein said structure stress definingequations include:provision for defining a set of displacement factorsto be imposed on a structure model; provision for calculating a loadvector representing externally applied forces applied to the structuremodel; and provision for modifying the load vector to compensate forrigid element deformation before calculating stresses on the structuremodel through use of the load vector; a processor for modeling thestructure and processing said structure displacement data through saidplurality of structure stress defining equations; a data output foroutputting said processed structure displacement data for structuralanalysis and modeling; and wherein said load vector is calculatedthrough the following equation:

     Pn!={Pn}+ Gm.sup.T !{Pm}+ K.sub.nm !{Ym}+ Gm.sup.T ! Kmm!{Ym}(9)

P_(n) ! is the load vector reduced to the independent set; {P_(n) } isthe partition of the load vector corresponding to the independent set;Gm^(T) ! is the transpose of G_(m) !, where G_(m) is defined as:-- R_(m)!⁻¹ R_(n) ! defined in Equation 4; {P_(m) } is the partition of the loadvector corresponding to the dependent set; K_(nm) ! represents thecouplings between the independent and dependent degrees of freedom;{Y_(m) } is-- R_(m) !⁻¹ {δ_(G) }; and K_(mm) ! is the partition of theglobal stiffness matrix pertaining to dependent degrees of freedom. 2.The system of claim 1, wherein said structure stress defining equationscomprise the following equation for recovering displacements:

    {u.sub.m }= R.sub.m !.sup.-1 {δ.sub.G }+ G.sub.m !{u.sub.n }(4)

Where: R_(m) !⁻¹ is the inverse of the partition of the dependentmulti-point constraint coefficient matrix {δ_(G) } is the user defineddisplacement term R_(n) !=is the partition of the multi-point constraintcoefficient matrix corresponding to the dependent set {u_(n) } is theindependent set of displacements.
 3. The system of claim 1 wherein saidstructure stress defining equations are adjusted to compensate for rigidelement mechanical expansion.
 4. The system of claim 1 wherein saidstructure stress defining equations are adjusted to compensate for rigidelement hydraulic expansion.
 5. The system of claim 1, wherein saidstructure stress defining equations include:provision for modifying therecovery of displacements to compensate for rigid element deformation.6. A system for analyzing characteristics of a structure having a rigidelement, comprising:a data input for inputting structure displacementdata; a memory for storing a plurality of structure stress definingequations adjusted to compensate for deformation of the rigid elementand including an equation to determine a load vector; a processor formodeling the structure and processing said structure displacement datathrough said plurality of structure stress defining equations; a dataoutput for outputting said processed structure displacement data forstructural analysis and modeling; and wherein said load vector iscalculated through the following equation:

     P.sub.n !={P.sub.n }+ G.sub.m.sup.T !{P.sub.m }+ K.sub.nm !{Y.sub.m }+ G.sub.m.sup.T ! K.sub.mm!{Y.sub.m }                    (9)

P_(n) ! is the load vector reduced to the independent set; {P_(n) } isthe partition of the load vector corresponding to the independent sets;G_(m) ^(T) ! is the transpose of G_(m) ! where G_(m) !, is defined as:--R_(m) !⁻¹ R_(n) ! as defined in Equation 4;. {P_(m) } is the partitionof the load vector corresponding to the dependent set; K_(nm) !represents the couplings between the independent and dependent degreesof freedom; {Y_(m) } is-- R_(m) !⁻¹ {δ_(G) }; and K_(mm) ! is thepartition of the global stiffness matrix pertaining to dependent degreesof freedom.